Maths Learning Service: Revision Mathematics IA
Matrices Mathematics IMA
A matrix is an array of numbers, written within a set of [ ] brackets, and arranged into a
pattern of rows and columns. For example:
" #
4 5 6 , 1 0 0 , 217−49
1 2 3 0 0 1h i
0 1 0
The order (or size, or dimension) of a matrix is written as “m ×n” where m = the number of
rows, and n = the number of columns. For example, the matrices above have dimensions
2 × 3, 3 × 3 and 1 × 4.
Basic Matrix Operations
Addition (or subtraction) of matrices is performed by adding (or subtracting) elements in
corresponding positions. Addition is only valid if the two matrices have the same order.
Examples:
(i)
"
1 − 3 5 # + "7 0 − 2 # =" 1+7 3+0
− −
5 + ( 2) # = " 6 3 − 3 #
2 4 0 3 4 1 2+3 4+4 0+( 1) 5 0 1
"− " − − −
(ii) 2 0 #− 81 # = " 2 −( 8) 0 − 1 # = " 6 −1 #
3 4 17 3 1 4 7 2 3
− − −−− − −
2 4 0 3 4
"
(iii) −1 − 3 5 # + " −2 0 # cannot be done as the orders are di fferent.
When a matrix is multiplied by a real number (called a scalar), each element is multiplied
by the scalar. The result is another matrix of the same order.
Examples:
(i) 4 3 9 = 4 × 3 4× 9 = 12 36
2 1 4 2 4 1 8 4
− 0 5 4 ×− 4× 5 − 0 20
− × ×− −
1
(ii) 2 h 7 8 −10 6 0.4 i = h 3.5 4 −5 # 3 0.2 i # #
" 0 6# − " 1 7
#
" 0 12 −
" 3 21 " 3 33
(iii) 2 5 −3 3 3 4 = 10 −6 9 12 = 1 −18
− − − − −
, Matrices 2007 Maths IA & IMA Revision/2
When giving matrices a name, use capital letters such as A, B, etc to distinguish them
from algebraic scalars such as a, b, etc.
Exercises
(1) Given that
−1
A= 2 0 B= 7 1 −3 C= 1 2 D= 11 5 E= 1 0
# 0 1
" 4 5 3# " 2 0 6# " −4 9 # " 0 −2 0 3
find the following (if possible):
(a) A+B (b) B+A (c) C+D (d) C−D (e) D − C
(f) A+E (g) B−D (h) 3A (i) 2C+D (j) 5B−4E
Matrix Multiplication
The rule for multiplying matrices can be represented as follows:
AB = row 1 of A col 1 of B row 1 of A col 2 of B row 1 of A col 3 of B ...
row 2 of A
×
×
col 1 of B row 2 of A
×
×
col 2 of B row 2 of A
×
×
col 3 of B ...
. . .
.
.
.
.
.
.
where “row i of A × col j of B” is a single number and stands for “each entry in row i of A is
multiplied by the corresponding entry in column j of B and the results are added together”.
Examples:
−
A= 1
"
4 #
5B= −1 −2 C= "3 4#
" #
D= 6 7 I=
"
0 1
#
2 1 3 4 −5
0 3
1 2 1 5 1 0
(i) CA =
" 3 4
#" 1 − 4 5
#
1 2 2 1 3
=
" ×
3 2+4× −
1 3 × ( 1) + 4 × 4 3 3+4 × 5
× #
=
"
10 13 29
#
1 2+2 11 (1)+2 41 3+2 5 4 713
× × ×− × × ×
(ii) AB = 2 −1 3 1 − 2
4 5
#
" 1 −
45 0 −3
=
" 1 ×
4+ − 4 × ( −
1)+5 × 0 1 × −
( 5) + − 4 × (
−
2)+5 × 3
# =
"0 2#
2 4+( 1) ( 1)+3 0 2 ( 5)+( 1) ( 2)+3 3 9 1
× ×− × ×− ×− ×
" 1 2 # 4 −5
(iii) CB= 3 4 −1 −2
0 3
This is not possible because there are fewer entries in the rows of C (two) than in
the columns of B (three).
Matrices Mathematics IMA
A matrix is an array of numbers, written within a set of [ ] brackets, and arranged into a
pattern of rows and columns. For example:
" #
4 5 6 , 1 0 0 , 217−49
1 2 3 0 0 1h i
0 1 0
The order (or size, or dimension) of a matrix is written as “m ×n” where m = the number of
rows, and n = the number of columns. For example, the matrices above have dimensions
2 × 3, 3 × 3 and 1 × 4.
Basic Matrix Operations
Addition (or subtraction) of matrices is performed by adding (or subtracting) elements in
corresponding positions. Addition is only valid if the two matrices have the same order.
Examples:
(i)
"
1 − 3 5 # + "7 0 − 2 # =" 1+7 3+0
− −
5 + ( 2) # = " 6 3 − 3 #
2 4 0 3 4 1 2+3 4+4 0+( 1) 5 0 1
"− " − − −
(ii) 2 0 #− 81 # = " 2 −( 8) 0 − 1 # = " 6 −1 #
3 4 17 3 1 4 7 2 3
− − −−− − −
2 4 0 3 4
"
(iii) −1 − 3 5 # + " −2 0 # cannot be done as the orders are di fferent.
When a matrix is multiplied by a real number (called a scalar), each element is multiplied
by the scalar. The result is another matrix of the same order.
Examples:
(i) 4 3 9 = 4 × 3 4× 9 = 12 36
2 1 4 2 4 1 8 4
− 0 5 4 ×− 4× 5 − 0 20
− × ×− −
1
(ii) 2 h 7 8 −10 6 0.4 i = h 3.5 4 −5 # 3 0.2 i # #
" 0 6# − " 1 7
#
" 0 12 −
" 3 21 " 3 33
(iii) 2 5 −3 3 3 4 = 10 −6 9 12 = 1 −18
− − − − −
, Matrices 2007 Maths IA & IMA Revision/2
When giving matrices a name, use capital letters such as A, B, etc to distinguish them
from algebraic scalars such as a, b, etc.
Exercises
(1) Given that
−1
A= 2 0 B= 7 1 −3 C= 1 2 D= 11 5 E= 1 0
# 0 1
" 4 5 3# " 2 0 6# " −4 9 # " 0 −2 0 3
find the following (if possible):
(a) A+B (b) B+A (c) C+D (d) C−D (e) D − C
(f) A+E (g) B−D (h) 3A (i) 2C+D (j) 5B−4E
Matrix Multiplication
The rule for multiplying matrices can be represented as follows:
AB = row 1 of A col 1 of B row 1 of A col 2 of B row 1 of A col 3 of B ...
row 2 of A
×
×
col 1 of B row 2 of A
×
×
col 2 of B row 2 of A
×
×
col 3 of B ...
. . .
.
.
.
.
.
.
where “row i of A × col j of B” is a single number and stands for “each entry in row i of A is
multiplied by the corresponding entry in column j of B and the results are added together”.
Examples:
−
A= 1
"
4 #
5B= −1 −2 C= "3 4#
" #
D= 6 7 I=
"
0 1
#
2 1 3 4 −5
0 3
1 2 1 5 1 0
(i) CA =
" 3 4
#" 1 − 4 5
#
1 2 2 1 3
=
" ×
3 2+4× −
1 3 × ( 1) + 4 × 4 3 3+4 × 5
× #
=
"
10 13 29
#
1 2+2 11 (1)+2 41 3+2 5 4 713
× × ×− × × ×
(ii) AB = 2 −1 3 1 − 2
4 5
#
" 1 −
45 0 −3
=
" 1 ×
4+ − 4 × ( −
1)+5 × 0 1 × −
( 5) + − 4 × (
−
2)+5 × 3
# =
"0 2#
2 4+( 1) ( 1)+3 0 2 ( 5)+( 1) ( 2)+3 3 9 1
× ×− × ×− ×− ×
" 1 2 # 4 −5
(iii) CB= 3 4 −1 −2
0 3
This is not possible because there are fewer entries in the rows of C (two) than in
the columns of B (three).
